
UQLAB user manual
2.1.1.2 Special cases of univariate distributions
Most of the available (built-in) distributions can be defined similarly to the way a Gaussian
distribution was defined in Section 2.1.1.1, i.e., either by defining the two parameters of the
distribution or its moments (mean and standand deviation). The meaning of the parameters
is as described in Appendix A. Some special cases are the following:
• For the exponential distribution only one parameter (λ) exists. When such a distribution
is defined by its parameters only one element is needed (λ). Additionally when it is
defined by its moments only one element is needed again which corresponds to the
mean and standard deviation (that are equal).
• For the beta distribution there is the possibility of using four parameters when a custom
support [a, b] needs to be defined. For example, in order to define an element of an
input vector that follows a beta distribution with parameters [r, s] = [1, 2] and support
[a, b] = [0.5, 1.5] we do the following:
Input.Marginals.Type = 'Beta';
Input.Marginals.Parameters = [1, 2, 0.5, 1.5];
Similarly, we can define a beta distribution with moments [µ, σ] = [0.8, 0.2] and support
[a, b] = [0.5, 1.5] as follows:
Input.Marginals.Type = 'Beta';
Input.Marginals.Moments = [0.8, 0.2, 0.5, 1.5];
In this case the two parameters r, s are computed according to the equations in Sec-
tion A.9.
• A marginal obtained through kernel smoothing on a data set X can be represented by
Input.Marginals.Type = 'KS';
Input.Marginals.Parameters = X;
which by default uses a Gaussian kernel with bandwidth optimized on the data. The
additional subfield .Options can be used to optionally specify the kernel type, band-
width, and the distribution bounds. For instance, the additional lines of code
Input.Marginals.Options.Bandwidth = 0.1;
Input.Marginals.Options.Bounds = [0, 5];
specify that a kernel bandwidth w = 0.1 must be used, and that the KDE distribution
must be truncated within the interval [0, 5].
2.1.1.3 Marginals inferred from data
The code needed to instruct UQLAB to infer the marginals from data is explained in detail in
the companion UQLAB User Manual – Statistical inference.
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